1. The problem statement, all variables and given/known data This problem took me a lot of time
if [itex]\displaystyle g(x) = \lim_{y\,\to\, x} {f(y)}
[/itex] exist for any x, then g is continuous.2. Relevant equations3. The attempt at a solution
[itex]\displaystyle
\lim_{x\rightarrow a^+} {f(x)} = g(a)
[/itex], so if ##\displaystyle \epsilon > 0 ## then there is an ##\delta_1 > 0## such that for all x, if ##\displaystyle a < x < \delta_1 + a ## then ## |f(x) - g(a)| < \epsilon/2 ##.

For any of those x, ##\displaystyle \lim_{y\rightarrow x^-} f(x)## exist, so, there is a ##\displaystyle \delta_2 >0## such that for all y, if ## y < x < \delta_2 + y ##, then ## |f(y) - g(x)| < \delta/2 ##.